All Questions
14 questions
1
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0
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44
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Catalog of integral symmetric matrices
Let $g$ be an integral symmetric matrix (perhaps with even diagonal components), and define an equivalence relation $g\sim g'$ if $g=Ug'U^T$ with $U$ a unimodular integral matrix.
For fixed $\det g$ (...
2
votes
0
answers
90
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decidability special case of column generation problem
I have the following problem:
Input: sub-spaces $V_1, \dots, V_d$ of $\mathbb{Z}^{d}$
Question: are there $v_i \in V_i$ such that the matrix $(v_1, \dots, v_d)$ has determinant $\pm 1$ (equivalently, ...
- 55
8
votes
0
answers
335
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How to check two matrices for similitude over $\mathbb{Z}$?
General question. Let $A$ and $B$ be two $n\times n$-matrices over
$\mathbb{Z}$. How do I algorithmically check whether $A$ and $B$ are similar
(i.e., conjugate in the ring $\mathbb{Z}^{n\times n}$)?
...
- 33.8k
5
votes
3
answers
673
views
Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix
Let $M $ be a matrix in $ \operatorname{GL}(2, \mathbb{Z})$ that has at least one eigenvalue of absolute value strictly bigger than $1$. What are the finite index subgroups $H$ of $\mathbb{Z}^2$ such ...
- 823
3
votes
1
answer
302
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Deciding isometry of unimodular lattices by Gram matrices
Say I have two unimodular lattices $A$ and $B$, represented by their Gram matrices.
Question: Is there an algorithm to decide whether $A$ and $B$ are isometric, i.e. whether there exists a matrix $S \...
- 9,501
8
votes
1
answer
275
views
An integral Jacobson-Morozov theorem?
$\DeclareMathOperator\SL{SL}$I want to ask if there exists a version of the Jacobson–Morozov theorem for integer matrices. A first approximation would ask: given an integral unipotent matrix $m \in \...
- 2,242
2
votes
0
answers
122
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Distribution of Smith normal forms for lower triangular matrices with given diagonals
Given integers $m$ and $n$ and $d_1, \ldots, d_m \in \mathbb{Z}/n \mathbb{Z}$, consider the set of all lower-triangular matrices of dimension $m$ with diagonal elements equal to $d_i$. What can be ...
- 75
13
votes
2
answers
697
views
in search of a transformation between determinants
Motivated by this MO question. Consider the two matrices $A_n$ and $B_n$ with entries $\binom{2j}i$ and $\binom{n+1}{2j-i}$, respectively; for $1\leq i, \,j\leq n$.
I can show $\det A_n=\det B_n=2^{\...
- 43.2k
2
votes
0
answers
92
views
On design of a (preferrably unimodular) matrix
Assume each entry is in $\Bbb Z$.
Say we want to solve $Ax=b$ where known $A$ is $n\times n$, unknown $x$ is $n\times1$ and $b$ is $n\times1$.
The absolute value of minors of augmented matrix $[A|b]$...
- 13.9k
3
votes
1
answer
240
views
Unique upper triangular basis matrix of sublattice $\Lambda \subseteq \mathbb{Z}^n$
Let $\Lambda \subseteq \mathbb{Z}^n$ be a full rank sublattice. We find an upper triangular basis matrix $B \in \mathfrak{ut}(\mathbb{Z},n)$ of $\Lambda$. Is $B$ unique up to the right action of $\...
- 1,813
6
votes
3
answers
425
views
Probability a random matrix contains a short integer vector in its kernel
Consider a random $m$ by $n$ matrix $M$ with $m \leq n$, chosen uniformly over all those whose elements are in $\{0,1\}$ (or $\{-1,1\}$ if it is any easier). Is there any mathematical theory that ...
- 3,377
1
vote
0
answers
60
views
Finding special vectors generated by a matrix
Let $G\in \Bbb Z^{n\times n}$ be a unimodular matrix.
Are there any efficient algorithms to find the maximum norm of a vector $v$ that satisfies $\langle\Delta(v),v\rangle=0$ over all vectors $v\in ...
- 13.9k
1
vote
0
answers
158
views
Comparing the volume of a rational lagrangian under a linear symplectomorphism.
Let's fix the standard symplectic structure $(\mathbb{R}^{2g}, \omega, J)$. A (marked) symplectic lattice then has the form $A\mathbb{Z}^{2g}$ for $A \in Sp_{2g}\mathbb{R}$. We say a vector subspace $...
5
votes
3
answers
1k
views
Matrices generating non-discrete subgroups of SL(2,R)
Jorgensen's inequality $\mid \left(Tr\left(A\right)\right)^2-4\mid+\mid Tr\left[A,B\right]-2\mid\ge 1$ gives a necessary condition for two matrices A,B to generate a discrete subgroup of SL(2,R). Are ...
- 10.4k